Investigating the Ability of PINNs To Solve Burgers' PDE Near Finite-Time BlowUp

TL;DR

This paper investigates the ability of Physics Informed Neural Networks (PINNs) to accurately solve Burgers' PDE near finite-time blow-up, providing theoretical bounds and experimental validation of their stability and accuracy.

Contribution

The work offers the first rigorous theoretical analysis of PINNs' stability near PDE blow-ups and correlates bounds with empirical performance on Burgers' PDE.

Findings

PINNs can approximate solutions close to blow-up with quantifiable bounds.

Theoretical bounds correlate with empirical errors in PINN solutions.

PINNs demonstrate potential in detecting finite-time singularities in PDEs.

Abstract

Physics Informed Neural Networks (PINNs) have been achieving ever newer feats of solving complicated PDEs numerically while offering an attractive trade-off between accuracy and speed of inference. A particularly challenging aspect of PDEs is that there exist simple PDEs which can evolve into singular solutions in finite time starting from smooth initial conditions. In recent times some striking experiments have suggested that PINNs might be good at even detecting such finite-time blow-ups. In this work, we embark on a program to investigate this stability of PINNs from a rigorous theoretical viewpoint. Firstly, we derive generalization bounds for PINNs for Burgers' PDE, in arbitrary dimensions, under conditions that allow for a finite-time blow-up. Then we demonstrate via experiments that our bounds are significantly correlated to the $\ell_2$-distance of the neurally found surrogate from the true blow-up solution, when computed on sequences of PDEs that are getting increasingly close to a blow-up.